Mathematics for the interested outsider

Tangent Vectors and Coordinates

Let’s say we have a coordinate patch around a point in an -dimensional manifold . We can use the function to give us some tangent vectors at called the “coordinate vectors”.

We define the coordinate vector as follows: given a smooth function , we define

Okay, I know that that’s confusing. But all we mean is this: start with a function . We compose it with the inverse of the coordinate map to get , where is some open neighborhood of the point . Now we can take that th partial derivative of this function and evaluate it at the point .

The first thing we really should check is that it doesn’t matter which representative we pick. That is, if in some neighborhood of , do we get the same answer? Indeed, in that case in some neighborhood of , and so their partial derivatives are identical. Thus this operation only depends on the germ .

But is it a tangent vector? It’s easy to see that it’s a linear functional, so we just have to check that it’s a derivation at :

And so we have at least these vectors at each point . We can even tell that they much be distinct — and even linearly independent — since we can calculate

where is the th coordinate projection . But we know that is always and everywhere — it takes the value if and otherwise.

Thus takes a different value on than on all the other . Further, any linear combination of the for must take the value on , while takes the value ; we see that none of the coordinate vectors can be written as a linear combination of the rest, and conclude that the dimension of is at least .

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I’ve always been bothered by the really poor notation here. It’s very tedious,cumbersome, and confusing to wade through all the different senses of parentheses (function/operator application, grouping), operators, functions, and the sets and function spaces in which they live. I know that people like to draw a picture of all the parts living in M versus R^n or R, and that helps. Maybe typographically reminding the reader what space each symbol lives in would help (by typesetting the spaces for each letter on the line above, or using color). Just thinking out loud. Hell, I might right it up with type annotations a la Haskell just to make it clearer to myself lol.

Sorry, I’m a visual thinker, and it helps to see visually just how this definition carries over our knowledge of R^n to M by all the coordinate patches.

[…] vector space of tangent vectors at . Given a coordinate patch around , we’ve constructed coordinate vectors at , and shown that they’re linearly independent in . I say that they also span the space, […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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